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User blog:Gonz0TheGreatt/Generalized Array Notation for Any Function
In an earlier blog post (see here) I defined my own array notation GGAN. In this post I want to generalize GGAN so that it can make an extension of any function. For a given function f, the array notation f# is called the GGAN extension of f. For instance, the GGAN extension of factorial is !#, the GGAN extension of TREE(n) is TREE#, etc. Let f be any increasing function over the natural numbers. Define the critical point of f as the smallest positive integer c such that f© > c. From now on, c denotes the critical point of f. Basic notation: Here we define the array fn_2, ... . This is an array of iteration; each entry in the array iterates f. B1: If there is only one entry, evaluate the function at that entry. fn = f(n). B2: If the last entry in the array is a 0, return the second to last entry. fn, 0 = n B3: Otherwise, decrement the last entry and replace the second to last entry with the original array with the last entry cut off. fb, n = ff[#, b, n - 1] Some theorems: fn = f^n(a) . f1, 2 = f^{f(a)}(a) . Leading 1s can be removed: f1 = f#. Higher Separators: Each separator in GGAN can be used analogously in the GGAN extension of any function. The rules are as follows: Colon Notation: C1: If the last entry is a 0 and occurs after a :, it can be removed. f: 0 = f#. C2: If there is only one : and only one entry afterwards, place a c before the rest of the body, and decrement the last entry. (Recall that c is the critical point of f) f: n = f# : n - 1 C3: If the last entry is a 0 and it is after a comma, return the second to last entry. C4: If the last entry is greater than 0 and is after a comma, replace the second to last entry with the array with the last entry cut off, and decrement the last entry. C5: If there is more than one : and only one entry after the last :, place a c after the previous :. The rest of the rules are the same as the rules for regular GGAN. The main difference is that the GGAN extension of a function does not have a base. Therefore, instead of using the base, I use f© for everything above Colon Notation. For example, TREE:: 4 = TREE: 3 : 3 : 3 : 5 = TREE: 3 : 3 : 2, 2, 2, 2, 2, 3. This is because the critical point of TREE is 2, and TREE(2) = 3. One way to view my original GGAN arrays are as the GGAN extensions of the function f(n) = b \cdot n for the base b . As long as b > 1 (which will always be the case as long as our array is worth anything), the critical point of f is c = 1 , and f© = b . Thus, regular GGAN could be viewed as the GGAN extension of multiplication. Vertical Bar Notation: Similarly to Vertical Bar Notation for GGAN, the array f| # tells us to ignore the critical point of f and use n everywhere instead. Additionally, the array f(a) | # tells us to ignore the critical point of f at every separator whose degree is at least a. Category:Blog posts